Prove that the sequence
lim n-->infinity 1/n(1 + 1/2 + ...+ 1/n) = 0
is
(i) monotone
(ii) bounded
(iii) find its limits
How can i proceed to prove that it is monotone and bounded???
Help me please!
Thank You!
If 1 + 1/2 + ... + 1/n is in the denominator, then the denominator itself is increasing (at each step the sum gets bigger, and it gets multiplier by a greater number). This implies that the sequence is decreasing. It is bounded below since all its terms are positive. And it will converge to 0, since it is smaller than 1/n, which converges to 0.
$\displaystyle {a}_{n}=\frac{1}{n}\sum_{k=1}^{n}\frac{1}{k}
=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{n} }\frac{1}{k}
<\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{{k}^{3/2}}$
The last expression tends to zero, since it is the product of a convergence sequence (the series) and an infinitesimal one. This proves that a_n tends to zero
This is a subtopic of the problem of sequence of means.
Suppose that $\displaystyle (a_n\ge 0)$ is a sequence define $\displaystyle S_n = \sum\limits_{k = 1}^n {a_k } \,\& \,M_n = \frac{{S_n }}{n}$.
Theorem: If $\displaystyle (a_n)$ monotonic then $\displaystyle (M_n)$ is monotonic.
Theorem: If $\displaystyle (a_n)\to L$ then $\displaystyle (M_n)\to L$.